Finding Equivalent Equations Of A Straight Line

Hey guys! Let's dive into the fascinating world of linear equations! We're given a line that gracefully passes through two points: $(-4, 10)$ and $(-1, 5)$. The equation that describes this line is $y=-\frac{5}{3}(x-2)$. Our mission, should we choose to accept it, is to find other equations that represent the same line. This is like finding different outfits for the same person – they all look different, but they're still the same! We'll explore several options and see which ones fit the bill. This is a super important concept in mathematics because it highlights that a single line can be represented in multiple forms. Understanding this flexibility is key to mastering linear equations and, frankly, makes life a whole lot easier when solving problems. Now, let's get our math on and find those equivalent equations! Let's analyze the question given to determine the correct equation that matches the line equation. Remember, a straight line is defined by a consistent slope and y-intercept; therefore, any equations representing the same straight line will have the same slope and y-intercept values.

Before we jump into the options, let's refresh our memory on the slope-intercept form of a linear equation, which is $y = mx + b$, where 'm' is the slope, and 'b' is the y-intercept. The equation is given in point-slope form, which is $y - y1 = m(x - x1)$, to convert it into slope-intercept form, we need to do some algebraic manipulation to get it to match the $y = mx + b$ format.

The given equation is $y=-\frac{5}{3}(x-2)$. Let's distribute the $-\frac{5}{3}$: $y = -\frac{5}{3}x + \frac{10}{3}$. Now we have the equation in slope-intercept form. This tells us the slope (m) is $-\frac{5}{3}$ and the y-intercept (b) is $\frac{10}{3}$. We will keep this in mind as we evaluate the answer options. So, let us see which options are equivalent to the original equation.

Decoding the Equations: Option by Option

Alright, let's get our detective hats on and examine each option to see if it represents the same line. Remember, our target is the slope-intercept form: $y = -\frac{5}{3}x + \frac{10}{3}$. We'll check the given options one by one, and if they have the same slope and y-intercept, they're in! Let's go!

Option A: $y=-\frac{5}{3} x-2$

Looking at this equation, it's already in slope-intercept form. We can immediately see that the slope is $-\frac{5}{3}$, which matches our original equation. However, the y-intercept is -2, which does not match our original equation's y-intercept of $\frac{10}{3}$. Therefore, option A is a no-go.

Option B: $y=-\frac{5}{3} x+\frac{10}{3}$

This one is also in slope-intercept form. The slope is $-\frac{5}{3}$, which is a match! And the y-intercept is $\frac{10}{3}$, which also matches our original equation. Bingo! Option B represents the same line, so we're marking this one as a correct answer. It's like finding a twin!

Option C: $3y=-5x+10$

This equation is not in slope-intercept form yet, so we need to do a little algebraic work. To get y by itself, we'll divide every term by 3: $y = -\frac{5}{3}x + \frac{10}{3}$. Hey, this is our original equation! Both the slope and y-intercept match. Therefore, option C is another correct answer. Nice work, team! We are on a roll now.

These equations represent the same line and are just different forms of expressing the same relationship between x and y. They are all mathematically equivalent.

Summarizing the Equivalent Equations

So, to recap, we started with the equation $y=-\frac{5}{3}(x-2)$ which is in point-slope form, and we found the following equations that also represent the same line:

• Option B: $y=-\frac{5}{3} x+\frac{10}{3}$
• Option C: $3 y=-5 x+10$

These are all different forms, but they represent the same line because they have the same slope and y-intercept when converted to slope-intercept form. Understanding equivalent equations is a powerful skill, and now you've got it! Keep practicing, and you'll become a linear equation master in no time! Remember that you can also work backward. If you are given the equation in slope-intercept form, you can convert it to point-slope form by simply manipulating the equation with algebra. The key thing is to ensure that the slope and y-intercept values remain the same. This whole idea is about recognizing that math, in many cases, offers flexibility in how we represent things.

Final Thoughts: Why This Matters

Why does this even matter, right? Well, understanding equivalent equations is like having multiple tools in your toolbox. When solving problems, you might find that one form of an equation is easier to work with than another. Being able to recognize and convert between different forms gives you that flexibility. It's also super helpful in more advanced math, like calculus, where you'll need to manipulate equations to solve complex problems. Furthermore, it is a great skill that provides a strong foundation for future topics. The ability to manipulate and understand different forms of linear equations is an essential skill to develop. So, keep practicing, and you'll find that these skills will come in handy as you tackle more advanced math topics! You're doing great, and keep up the fantastic work!